I am confused about the different notions for a closed smooth manifold to be oriented. In my mind there are several equivalent ones:
1) Coherent pointwise orientation of the tangent spaces.
2) Choice of nonvanishing top dimensional form.
3) Orientation of the top dimensional de Rham cohomology group as a real vector space.
I understand the equivalence of these above notions. There are also some similar definitions for an orientation of a topological manifold:
1) Coherent choice of generators for the pointwise homology groups $H_n(M, M \setminus \{x\}; \mathbb{Z})$.
2) Choice of generator ("fundamental class") for the top homology group $H_n(M; \mathbb{Z})$.
My question is the following: how does a choice of smooth orientation induce a choice of topological orientation?
Thanks!